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%I A016789 #272 Apr 17 2024 11:15:01
%S A016789 2,5,8,11,14,17,20,23,26,29,32,35,38,41,44,47,50,53,56,59,62,65,68,71,
%T A016789 74,77,80,83,86,89,92,95,98,101,104,107,110,113,116,119,122,125,128,
%U A016789 131,134,137,140,143,146,149,152,155,158,161,164,167,170,173,176,179
%N A016789 a(n) = 3*n + 2.
%C A016789 Except for 1, n such that Sum_{k=1..n} (k mod 3)*binomial(n,k) is a power of 2. - _Benoit Cloitre_, Oct 17 2002
%C A016789 The sequence 0,0,2,0,0,5,0,0,8,... has a(n) = n*(1 + cos(2*Pi*n/3 + Pi/3) - sqrt(3)*sin(2*Pi*n + Pi/3))/3 and o.g.f. x^2(2+x^3)/(1-x^3)^2. - _Paul Barry_, Jan 28 2004 [_Artur Jasinski_, Dec 11 2007, remarks that this should read (3*n + 2)*(1 + cos(2*Pi*(3*n + 2)/3 + Pi/3) - sqrt(3)*sin(2*Pi*(3*n + 2)/3 + Pi/3))/3.]
%C A016789 Except for 2, exponents e such that x^e + x + 1 is reducible. - _N. J. A. Sloane_, Jul 19 2005
%C A016789 The trajectory of these numbers under iteration of sum of cubes of digits eventually turns out to be 371 or 407 (47 is the first of the second kind). - Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 19 2009
%C A016789 Union of A165334 and A165335. - _Reinhard Zumkeller_, Sep 17 2009
%C A016789 a(n) is the set of numbers congruent to {2,5,8} mod 9. - _Gary Detlefs_, Mar 07 2010
%C A016789 It appears that a(n) is the set of all values of y such that y^3 = k*n + 2 for integer k. - _Gary Detlefs_, Mar 08 2010
%C A016789 These numbers do not occur in A000217 (triangular numbers). - _Arkadiusz Wesolowski_, Jan 08 2012
%C A016789 A089911(2*a(n)) = 9. - _Reinhard Zumkeller_, Jul 05 2013
%C A016789 Also indices of even Bell numbers (A000110). - _Enrique Pérez Herrero_, Sep 10 2013
%C A016789 Central terms of the triangle A108872. - _Reinhard Zumkeller_, Oct 01 2014
%C A016789 A092942(a(n)) = 1 for n > 0. - _Reinhard Zumkeller_, Dec 13 2014
%C A016789 a(n-1), n >= 1, is also the complex dimension of the manifold E(S), the set of all second-order irreducible Fuchsian differential equations defined on P^1 = C U {oo}, having singular points at most in S = {a_1, ..., a_n, a_{n+1} = oo}, a subset of P^1. See the Iwasaki et al. reference, Proposition 2.1.3., p. 149. - _Wolfdieter Lang_, Apr 22 2016
%C A016789 Except for 2, exponents for which 1 + x^(n-1) + x^n is reducible. - _Ron Knott_, Sep 16 2016
%C A016789 The reciprocal sum of 8 distinct items from this sequence can be made equal to 1, with these terms: 2, 5, 8, 14, 20, 35, 41, 1640. - _Jinyuan Wang_, Nov 16 2018
%C A016789 There are no positive integers x, y, z such that 1/a(x) = 1/a(y) + 1/a(z). - _Jinyuan Wang_, Dec 31 2018
%C A016789 As a set of positive integers, it is the set sum S + S where S is the set of numbers in A016777. - _Michael Somos_, May 27 2019
%C A016789 Interleaving of A016933 and A016969. - _Leo Tavares_, Nov 16 2021
%C A016789 Prepended with {1}, these are the denominators of the elements of the 3x+1 semigroup, the numerators being A005408 prepended with {2}. See Applegate and Lagarias link for more information. - _Paolo Xausa_, Nov 20 2021
%C A016789 This is also the maximum number of moves starting with n + 1 dots in the game of Sprouts. - _Douglas Boffey_, Aug 01 2022 [See the Wikipedia link. - _Wolfdieter Lang_, Sep 29 2022]
%C A016789 a(n-2) is the maximum sum of the span (or L(2,1)-labeling number) of a graph of order n and its complement. The extremal graphs are stars and their complements. For example, K_{1,2} has span 3, and K_2 has span 2. Thus a(3-1) = 5. - _Allan Bickle_, Apr 20 2023
%D A016789 K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé, Vieweg, 1991. p. 149.
%D A016789 Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269
%H A016789 G. C. Greubel, <a href="/A016789/b016789.txt">Table of n, a(n) for n = 0..10000</a>
%H A016789 D. Applegate and J. C. Lagarias, <a href="https://doi.org/10.1016/j.jnt.2005.06.010">The 3x+1 semigroup</a>, Journal of Number Theory, Vol. 177, Issue 1, March 2006, pp. 146-159; see also the <a href="https://arxiv.org/abs/math/0411140">arXiv version</a>, arXiv:math/0411140 [math.NT], 2004-2005.
%H A016789 H. Balakrishnan and N. Deo, <a href="http://www.cs.ucf.edu/~deo/deo/Radiocoloring.pdf">Parallel algorithm for radiocoloring a graph</a>, Congr. Numer. 160 (2003), 193-204.
%H A016789 Allan Bickle, <a href="https://doi.org/10.1016/j.disc.2023.113392">Extremal Decompositions for Nordhaus-Gaddum Theorems</a>, Discrete Math, 346 7 (2023), 113392.
%H A016789 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a> p. 9.
%H A016789 L. Euler, <a href="https://arxiv.org/abs/math/0411587">An observation on the sums of divisors</a>, arXiv:math/0411587 [math.HO], 2004-2009, p. 9.
%H A016789 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=937">Encyclopedia of Combinatorial Structures 937</a>
%H A016789 L. B. W. Jolley, <a href="https://archive.org/details/summationofserie00joll">Summation of Series</a>, Dover, 1961, p. 16
%H A016789 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A016789 Konrad Knopp, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM1954.0001.001">Theorie und Anwendung der unendlichen Reihen</a>, Berlin, J. Springer, 1922. (Original German edition of "Theory and Application of Infinite Series")
%H A016789 Fabian S. Reid, <a href="https://arxiv.org/abs/2105.07955">The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles</a>, arXiv:2105.07955 [math.GM], 2021.
%H A016789 Luis Manuel Rivera, <a href="https://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%H A016789 Leo Tavares, <a href="/A016789/a016789_1.jpg">Illustration: Capped Triangular Frames</a>
%H A016789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sprouts_(game)">Sprouts (game)</a>
%H A016789 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A016789 G.f.: (2+x)/(1-x)^2.
%F A016789 a(n) = 3 + a(n-1).
%F A016789 a(n) = 1 + A016777(n).
%F A016789 a(n) = A124388(n)/9.
%F A016789 a(n) = A125199(n+1,1). - _Reinhard Zumkeller_, Nov 24 2006
%F A016789 Sum_{n>=1} (-1)^n/a(n) = (1/3)*(Pi/sqrt(3) - log(2)). - _Benoit Cloitre_, Apr 05 2002
%F A016789 1/2 - 1/5 + 1/8 - 1/11 + ... = (1/3)*(Pi/sqrt(3) - log 2). [Jolley] - _Gary W. Adamson_, Dec 16 2006
%F A016789 Sum_{n>=0} 1/(a(2*n)*a(2*n+1)) = (Pi/sqrt(3) - log 2)/9 = 0.12451569... (see A196548). [Jolley p. 48 eq (263)]
%F A016789 a(n) = 2*a(n-1) - a(n-2); a(0)=2, a(1)=5. - _Philippe Deléham_, Nov 03 2008
%F A016789 a(n) = 6*n - a(n-1) + 1 with a(0)=2. - _Vincenzo Librandi_, Aug 25 2010
%F A016789 Conjecture: a(n) = n XOR A005351(n+1) XOR A005352(n+1). - Gilian Breysens, Jul 21 2017
%F A016789 E.g.f.: (2 + 3*x)*exp(x). - _G. C. Greubel_, Nov 02 2018
%F A016789 a(n) = A005449(n+1) - A005449(n). - _Jinyuan Wang_, Feb 03 2019
%F A016789 a(n) = -A016777(-1-n) for all n in Z. - _Michael Somos_, May 27 2019
%F A016789 a(n) = A007310(n+1) + (1 - n mod 2). - _Walt Rorie-Baety_, Sep 13 2021
%F A016789 a(n) = A000096(n+1) - A000217(n-1). See Capped Triangular Frames illustration. - _Leo Tavares_, Oct 05 2021
%e A016789 G.f. = 2 + 5*x + 8*x^2 + 11*x^3 + 14*x^4 + 17*x^5 + 20*x^6 + ... - _Michael Somos_, May 27 2019
%p A016789 seq(3*n+2, n = 0 .. 50); # _Matt C. Anderson_, May 18 2017
%t A016789 Range[2, 500, 3] (* _Vladimir Joseph Stephan Orlovsky_, May 26 2011 *)
%t A016789 LinearRecurrence[{2,-1},{2,5},70] (* _Harvey P. Dale_, Aug 11 2021 *)
%o A016789 (Haskell)
%o A016789 a016789 = (+ 2) . (* 3)  -- _Reinhard Zumkeller_, Jul 05 2013
%o A016789 (PARI) vector(100,n,3*n-1) \\ _Derek Orr_, Apr 13 2015
%o A016789 (Magma) [3*n+2: n in [0..80]]; // _Vincenzo Librandi_, Apr 14 2015
%o A016789 (GAP) List([0..70],n->3*n+2); # _Muniru A Asiru_, Nov 02 2018
%o A016789 (Python) for n in range(0,100): print(3*n+2, end=', ') # _Stefano Spezia_, Nov 21 2018
%Y A016789 First differences of A005449.
%Y A016789 Cf. A002939, A017041, A017485, A125202, A017233, A179896, A017617, A016957, A008544 (partial products), A032766, A016777, A124388, A005351.
%Y A016789 Cf. A087370.
%Y A016789 Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.
%Y A016789 Cf. A000096, A000217.
%Y A016789 Cf. A016933, A016969.
%K A016789 nonn,easy
%O A016789 0,1
%A A016789 _N. J. A. Sloane_

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